Algorithmica最新文献

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seems to escape from the literature. A path containing at least k vertices is considered long. When (k le 3), the problem is polynomial time solvable; when k is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed (k ge 4), the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a k-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when (k = 4), the problem admits a 4-approximation algorithm which was presented recently. We propose the first ((0.4394 k + O(1)))-approximation algorithm for the general problem and an improved 2-approximation algorithm when (k = 4). Both algorithms are based on local improvement, and their theoretical performance analyses are done via amortization and their practical performance is examined through simulation studies.

Given two permutations, a pattern (sigma ) and a text (pi ), Parity Permutation Pattern Matching asks whether there exists a parity and order preserving embedding of (sigma ) into (pi ). While it is known that Permutation Pattern Matching is in (textsc {FPT}), we show that adding the parity constraint to the problem makes it (textsc {W}[1])-hard, even for alternating permutations or for 4321-avoiding patterns. However, the problem remains in (textsc {FPT}) if (pi ) avoids a fixed permutation, thanks to a recent meta-theorem on twin-width. On the other hand, as for the classical version, Parity Permutation Pattern Matching remains polynomial-time solvable when the pattern is separable, or if both permutations are 321-avoiding, but NP-hard if (sigma ) is 321-avoiding and (pi ) is 4321-avoiding.

We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Williamson et al. in Combinatorica 15(3):435–454, 1995. https://doi.org/10.1007/BF01299747). Williamson et al. prove an approximation ratio of two for connectivity augmentation problems where the connectivity requirements can be specified by uncrossable functions. They state: “Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar ... A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems.” Our main result proves that the primal-dual algorithm of Williamson et al. achieves an approximation ratio of (16) for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions has a laminar support. We present three applications of our main result to problems in the area of network design. (1)  A (16)-approximation algorithm for augmenting a family of small cuts of a graph G. The previous best approximation ratio was (O(log {|V(G)|})). (2)  A (16cdot {lceil k/u_{min} rceil })-approximation algorithm for the Cap-k-ECSS problem which is as follows: Given an undirected graph (G = (V,E)) with edge costs (c in {mathbb {Q}}_{ge 0}^E) and edge capacities (u in {mathbb {Z}}_{ge 0}^E), find a minimum-cost subset of the edges (Fsubseteq E) such that the capacity of any cut in (V, F) is at least k; (u_{min}) (respectively, (u_{max})) denotes the minimum (respectively, maximum) capacity of an edge in E, and w.l.o.g. (u_{max} le k). The previous best approximation ratio was (min (O(log {|V|}), k, 2u_{max})). (3)  A (20)-approximation algorithm for the model of (p, 2)-Flexible Graph Connectivity. The previous best approximation ratio was (O(log {|V(G)|})), where G denotes the input graph.

k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an (O(nlog k)+(1/epsilon )^{O(2^dk^{1-1/d}log k)})-time ((1+epsilon ))-approximation for Euclidean k-center, where d is the dimension. We show for a closely related problem, k-supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k-center and k-supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a (2^{O(klog k)}n^2) time 3-approximation for NUkC in general metrics, and a (2^{O((klog k)/epsilon )}dn) time ((1+epsilon ))-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.

In this work, we study the problem of approximating the distance to subsequence-freeness in the sample-based distribution-free model. For a given subsequence (word) (w = w_1 ldots w_k), a sequence (text) (T = t_1 ldots t_n) is said to contain w if there exist indices (1 le i_1< cdots < i_k le n) such that (t_{i_{j}} = w_j) for every (1 le j le k). Otherwise, T is w-free. Ron and Rosin (ACM Trans Comput Theory 14(4):1–31, 2022) showed that the number of samples both necessary and sufficient for one-sided error testing of subsequence-freeness in the sample-based distribution-free model is (Theta (k/epsilon )). Denoting by (Delta (T,w,p)) the distance of T to w-freeness under a distribution (p:[n]rightarrow [0,1]), we are interested in obtaining an estimate (widehat{Delta }), such that (|widehat{Delta }- Delta (T,w,p)| le delta ) with probability at least 2/3, for a given error parameter (delta ). Our main result is a sample-based distribution-free algorithm whose sample complexity is (tilde{O}(k^2/delta ^2)). We first present an algorithm that works when the underlying distribution p is uniform, and then show how it can be modified to work for any (unknown) distribution p. We also show that a quadratic dependence on (1/delta ) is necessary.

We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the ((1+1)) evolutionary algorithm on the Needle problem due to Garnier et al. (Evol Comput 7:173–203, 1999). We also use this method to analyze the runtime of the ((1+1)) evolutionary algorithm on a benchmark consisting of (n/ell ) plateaus of effective size (2^ell -1) which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For (ell = o(n)), the optimal static mutation rate is approximately 1.59/n. The optimal fitness dependent mutation rate, when the first k fitness-relevant bits have been found, is asymptotically (1/(k+1)). These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that the Fourier analysis approach can be applied to other plateau problems as well.

The planar slope number ({{,textrm{psn},}}(G)) of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that ({{,textrm{psn},}}(G) in O(c^{Delta })) for every planar graph G of maximum degree (Delta ). This upper bound has been improved to (O(Delta ^5)) if G has treewidth three, and to (O(Delta )) if G has treewidth two. In this paper we prove ({{,textrm{psn},}}(G) le max {4,Delta }) when G is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that (O(Delta ^2)) slopes suffice for nested pseudotrees.

Based on well-known complexity theory conjectures, any polynomial-time kernelization algorithm for the NP-hard Line-Cover problem produces a kernel of size (Omega (k^2)), where k is the size of the sought line cover. Motivated by the current research in massive data processing, we study the existence of kernelization algorithms with limited space and time complexity for Line-Cover. We prove that every kernelization algorithm for Line-Cover takes time (Omega (n log k + k^2 log k)), and present a randomized kernelization algorithm for Line-Cover that produces a kernel of size bounded by (k^2), and runs in time ({mathcal {O}}(n log k + k^2 (log k log log k)^2)) and space ({mathcal {O}}(k^2log ^{2} k)). Our techniques are also useful for developing deterministic kernelization algorithms for Line-Cover with limited space and improved running time, and for developing streaming kernelization algorithms for Line-Cover with near-optimal update-time.

We study the Online Multiset Submodular Cover problem (OMSC), where we are given a universe U of elements and a collection of subsets (mathcal {S}subseteq 2^U). Each element (u_j in U) is associated with a nonnegative, nondecreasing, submodular polynomially computable set function (f_j). Initially, the elements are uncovered, and therefore we pay a penalty per each unit of uncovered element. Subsets with various coverage and cost arrive online. Upon arrival of a new subset, the online algorithm must decide how many copies of the arriving subset to add to the solution. This decision is irrevocable, in the sense that the algorithm will not be able to add more copies of this subset in the future. On the other hand, the algorithm can drop copies of a subset, but such copies cannot be retrieved later. The goal is to minimize the total cost of subsets taken plus penalties for uncovered elements. We present an (O(sqrt{rho _{max }}))-competitive algorithm for OMSC that does not dismiss subset copies that were taken into the solution, but relies on prior knowledge of the value of (rho _{max }), where (rho _{max }) is the maximum ratio, over all subsets, between the penalties covered by a subset and its cost. We provide an (Oleft( log (rho _{max }) sqrt{rho _{max }} right) )-competitive algorithm for OMSC that does not rely on advance knowledge of (rho _{max }) but uses dismissals of previously taken subsets. Finally, for the capacitated versions of the Online Multiset Multicover problem, we obtain an (O(sqrt{rho _{max }'}))-competitive algorithm when (rho _{max }') is known and an (Oleft( log (rho _{max }') sqrt{rho _{max }'} right) )-competitive algorithm when (rho _{max }') is unknown, where (rho _{max }') is the maximum ratio over all subset incarnations between the penalties covered by this incarnation and its cost.

Co-evolutionary algorithms have a wide range of applications, such as in hardware design, evolution of strategies for board games, and patching software bugs. However, these algorithms are poorly understood and applications are often limited by pathological behaviour, such as loss of gradient, relative over-generalisation, and mediocre objective stasis. It is an open challenge to develop a theory that can predict when co-evolutionary algorithms find solutions efficiently and reliable. This paper provides a first step in developing runtime analysis for population-based competitive co-evolutionary algorithms. We provide a mathematical framework for describing and reasoning about the performance of co-evolutionary processes. To illustrate the framework, we introduce a population-based co-evolutionary algorithm called PDCoEA, and prove that it obtains a solution to a bilinear maximin optimisation problem in expected polynomial time. Finally, we describe settings where PDCoEA needs exponential time with overwhelmingly high probability to obtain a solution.